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Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

Yuxiang Ji, Yanir A. Rubinstein, Kewei Zhang

Abstract

Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hanson space and Calabi's Ricci flat spaces as limits of compact singular Einstein spaces.

Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

Abstract

Calabi--Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kahler--Einstein edge metrics singular along two disjoint divisors on the Calabi--Hirzebruch manifolds and study their Gromov--Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hanson space and Calabi's Ricci flat spaces as limits of compact singular Einstein spaces.

Paper Structure

This paper contains 19 sections, 34 theorems, 197 equations, 2 figures, 1 table.

Table of Contents

  1. Motivation
  2. Results
  3. Large angle limits
  4. Small angle limits
  5. Organization
  6. Acknowledgments.
  7. Model metrics
  8. Non-compact case: model metrics on $-kH_{\mathbb{P}^{n-1}}$
  9. Compact case: model metrics on $\mathbb{P}^{n}(1,\dots, 1,k)$
  10. Constructions of Kähler--Einstein edge metrics
  11. Constructions of $\eta_{\beta_1}$ on $\mathbb{F}_{n, k}$
  12. Constructions of $\xi_{\beta_2}$ on $\mathbb{F}_{n, k}$
  13. Angle asymptotics
  14. Large angle limits
  15. Small angle limits and fiberwise rescaling
  16. ...and 4 more sections

Key Result

Theorem 2.4

Fix a base point $p$ on the zero section of $\mathbb{F}_{n, k}$ and $q$ on the infinity section. The pointed metric space $(\mathbb{F}_{n, k}, \eta_{\beta_1}, p)$ converges in the pointed Gromov--Hausdorff sense to $(-kH_{\mathbb{P}^{n-1}}, \omega_{\mathrm{eh}, n, k}, p)$ as $\beta_1$ tends to $n/k$

Figures (2)

  • Figure 1: The upper part shows the Kähler edge structure on the Calabi--Hirzebruch manifold $\mathbb{F}_{n, k}$. When $n=2$, $\mathbb{F}_{2,k}$ is the $k$-th Hirzebruch surface and $Z_{2,\pm k}$ is a $\mp k$-curve. The lower part shows the different limits described in Theorem \ref{['thm: main']}. Note that on the left, we get a non-compact limit, with $q$ (together with all of $Z_{n,-k}$) pushed-out to infinity. On the right we get a compact limit, with $p$ limiting (together with all of $Z_{n,k}$) to an isolated orbifold point.
  • Figure 2: Blow up of $\mathbb{P}^2(1,1,k)$ at $p$ (with $k>1$).

Theorems & Definitions (82)

  • Conjecture 1.1
  • Conjecture 1.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Definition 3.1
  • ...and 72 more